Abstract
In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it ‘Fourier–Jacobi type’, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).
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References
Berndt, R. and Schmidt, R.: Elements of the Representation Theory of the Jacobi Group, Progr. in Math. 163, Birkhäuser, Basel, 1998.
Corwin, L. and Greenleaf, F. P.: Representations of Nilpotent Lie Groups and their Applications Part 1: Basic Theory and Examples, Cambridge Stud. Adv. Math. 18 Cambridge University Press, 1990.
Eichler, M. and Zagier, D.: The Theory of Jacobi Forms, Progr. in Math. 55, Birkhäuser, Basel, 1985.
Erdélyi, A. et al.: Higher Transcendental Functions I, McGraw Hill, New York, 1953.
Gelbert, S. S.: Weil's Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Math. 530, Springer, New York, 1976.
Hirano, M.: Fourier-Jacobi type spherical functions on Sp(2, ℝ), Thesis, Univ. of Tokyo (1998).
Hirano, M.: Fourier Jacobi type spherical functions for PJ-principal series representations of Sp(2, ℝ), Preprint.
Knapp, A. W.: Representation Theory of Semisimple Groups; An Overview Based on Examples, Princeton Univ. Press, 1986.
Kostant, B.: On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101-184.
Mackey, G.: Unitary representations of group extensions 1, Acta Math. 99 (1958), 265-311.
Meijer, C. S.: On the G-function. I-VIII, Indag. Math. 8 (1946), 124-134, 213-225, 312-324, 391-400, 468-475, 595-602, 661-670, 713-723.
Murase, A. and Sugano, T.: Whittaker-Shintani functions on the symplectic group of Fourier Jacobi type, Compositio Math. 79 (1991), 321-349.
Oda, T.: An explicit integral representation of Whittaker functions on Sp(2, ℝ) for the large discrete series representations, Tôhoku Math. J. 46 (1994), 261-279.
Satake, I.: Unitary representations of a semi-direct product of Lie groups on \(\overline \partial \)-cohomology spaces, Math. Ann. 190 (1971), 177-202.
Shalika, J. A.: The multiplicity one theorem for GL n, Ann. of Math. 100 (1974), 171-193.
Vogan, D. A.: Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98.
Wallach, N.: Asymptotic expansions of generalized matrix entries of representations of real reductive groups, In: Lie Group Representation I, Lecture Notes in Math. 1024, Springer, New York, 1984, pp. 287-369.
Yamashita, H.: Finite multiplicity theorem for induced representations of semisimple Lie groups II — Applications to generalized Gelfand-Graev representations, J. Math. Kyoto Univ. 28 (1988), 383-444.
Yamashita, H.: Embeddings of discrete series into induced representations of semisimple Lie Groups I — General theory and the case of SU(2, 2), Japan J. Math. 16 (1990), 31-95.
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Hirano, M. Fourier–Jacobi Type Spherical Functions for Discrete Series Representations of Sp(2, R). Compositio Mathematica 128, 177–216 (2001). https://doi.org/10.1023/A:1017528120756
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DOI: https://doi.org/10.1023/A:1017528120756